Discrete invariants of varieties in positive characteristic. (English) Zbl 1084.14023

Moonen, Ben; Wedhorn, Torsten

Int. Math. Res. Not. 2004, No. 72, 3855-3903 (2004).

Let \(F: X \to S\) be a smooth proper morphism of sheaves in characteristic \(p >0\). If the sheaves \(R^b f_* \Omega^a_{X/S}\) are locally free and the Hodge-de Rham spectral sequence degenerates at \(E_1\), then the de Rham cohomology sheaves \(\mathcal M = H^m_{\text{dR}} (X/S)\) are locally free \(\mathcal O_S\)-modules equipped with a descending Hodge filtration \(C^\bullet\) and an ascending conjugate filtration \(D_\bullet\) as well as with \(\mathcal O_S\)-linear isomorphisms \(\varphi_i: (\text{gr}^i_C)^{(p)} \to \text{gr}_i^D\) given by the inverse Cartier operator. In this paper the authors give a complete classification of the structures \((\mathcal M, C^\bullet, D_\bullet, \varphi_\bullet)\) over an algebraically closed field. The result shows that such structures are essentially combinatorial objects.

Reviewer: Min Ho Lee (Cedar Falls)

MSC:

14F40	de Rham cohomology and algebraic geometry
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14G35	Modular and Shimura varieties
14J28	\(K3\) surfaces and Enriques surfaces

Keywords:

Hodge structures; Cartier operator; de Rham cohomology